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1 results

  • Flat bundles, von Neumann algebras and K-theory with ℝ/ℤ-coefficients

  • Paolo Antonini, Sara Azzali, Georges Skandalis
  • Journal:
    Journal of K-Theory / Volume 13 / Issue 2 / April 2014
    Published online by Cambridge University Press:
    24 February 2014, pp. 275-303
    Print publication:
    April 2014

  • Let M be a closed manifold and α: π1 (M) → Un a representation. We give a purely K-theoretic description of the associated element in the K-theory group of M with ℝ/ℤ-coefficients ([α] ∈ K1 (M; ℝ/ℤ)). To that end, it is convenient to describe the ℝ/ℤ-K-theory as a relative K-theory of the unital inclusion of ℂ into a finite von Neumann algebra B. We use the following fact: there is, associated with α, a finite von Neumann algebra B together with a flat bundle ℰ → M with fibers B, such that Eα ⊗ ℰ is canonically isomorphic with ℂn ⊗ ℰ, where Eα denotes the flat bundle with fiber ℂn associated with α. We also discuss the spectral flow and rho type description of the pairing of the class [α] with the K-homology class of an elliptic selfadjoint (pseudo)-differential operator D of order 1.